City were brighter after the half time break, and Gary Alexander had a great chance in the opening minute. He chased on to a long ball through the middle, but poked the ball wide with the Oxford 'keeper coming towards him.
Alexander and Richard Sneekes each had two chances but couldn't beat Woodman.
In the 75th minute, Theo Whitmore half-volleyed at goal from outside the area and Lee Philpott went close with the rebound after Woodman spilled the shot.
But City were made to rue these misses when a gap in our defence saw Brooks score from close range from Hackett's ball into the box.
City drop to 9th, but out of the six above us only Scunthorpe and Rushden won, so Tigers boss Billy Russell remains optimistic.
"I'm still hopeful that we'll make it but we have to change this mentality in the last ten minutes as it's costing us dearly.
"It is difficult because we're working extremely hard for very little reward at the present time but we have to keep going and believe in what we're trying to achieve."
City line-up (4-5-1): Musselwhite; Petty, Whittle, Greaves, Holt; Johnsson, Norris, Sneekes, Whitmore (Dudfield), Philpott (Beresford); Alexander.
Un-used subs: Glennon, Goodison, Matthews.